Folding Backstepping Approach to Parabolic PDE Bilateral Boundary Control

Abstract

We consider the stabilization problem for an unstable 1-D diffusion-reaction partial differential equation using a so-called folding transformation. The diffusion-reaction equation is transformed into a 2 ×2 system of coupled parabolic PDEs with exotic boundary conditions. A first backstepping transformation is designed to map the unstable system into a strict-feedback intermediate target system. A second backstepping transformation is designed to stabilize the intermediate target system. Interestingly, the companion gain kernel PDEs contain the folding boundary condition, exhibiting symmetry with the original system. The kernels posses a cascading structure that allows for sequential solution methods. Finally, the controller derived is shown to be exponentially stabilizing in the L2 sense.